The Research On Generalized Core Inverses In Rings With Involution

Moore-Penrose inverse and Drazin inverse are classical generalized inverses.They play important roles in many fields.With the development of the theory of generalized inverses,there appeared many new generalized inverses,such as the core inverse.Since the core inverse is restricted to matrices of index 1,as an extension of the core inverse,authors introduced generalized inverses for matrices of arbitrary indices,namely the core-EP inverse and the DMP inverse,collectively referred to as generalized core inverses.This thesis mainly focuses on the above introduced generalized core inverses in rings with involution.The main contents are arranged as follows:In Chapter 2,we mainly study the pseudo core inverse in a ring with involution.Firstly,the existence criteria and expressions of the pseudo core inverse are given.The most important work is that we characterize the core-EP inverse introduced by K.Manjunatha Prasad et al.as the unique solution of three equations,generalizing the core-EP inverse from complex matrices to an arbitrary ring,and we call it the pseudo core inverse.Then we reveal the relations amongthe pseudo core inverse,the(b,c)-inverse and generalized inverse AT,S(2).In addition,necessary and sufficient conditions under which the reverse order law and absorption law of the pseudo core inverse hold are given.Finally,the core-EP inverse of a complex matrix are obtained by Hartwig-Spindelbock decomposition and Jordan decomposition respectively.In Chapter 3,we mainly investigate the existence criteria and expressions for the pseudo core inverse of a matrix over a ring.First of all,the pseudo core inverse of a matrix product PAQ is considered under prescribed conditions,so that corresponding results for the core inverse are extended.As an application,we discuss the pseudo core invertibility of a lower triangular matrix by pseudo core inverses of diagonal elements.Finally,we calculate the pseudo core inverse of a companion matrix by using a {1,3}-inverse of a Toeplitz matrix.In Chapter 4,we consider*-DMP elements in a ring.Firstly,*-DMP elements are charac-terized in terms of the pseudo core inverse.It is proved that a is a*-DMP element if and only if the pseudo core inverse of a exists and commutes with a.Secondly,the core-EP decomposition and core-EP order proposed by H.X.Wang are extended from complex matrices to rings by using pure algebraic techniques so that more characterizations of*-DMP elements are given.Finally,the necessary and sufficient conditions for a class of specific matrices over a ring R to be*-DMP matrices are obtained.When R is a semi-simple Artinian ring,these matrices include all square matrices over R.Chapter 5 devotes to the study of the W-weighted core-EP inverse of a complex rect-angular matrix.Firstly,the W-weighted core-EP inverse is characterized in terms of three equations so that the accuracy of a given calculation method can be measured by the resid-ual norm.Then,computational expressions of the W-weighted core-EP inverse are obtained by singular value decomposition,full rank decomposition and QR decomposition respective-ly.Secondly,the relationship between the weighted core-EP reverse and the W-weighted Drazin reverse are revealed.Finally,we define and characterize a new binary relation on the set of rectangular complex matrices involving the W-weighted core-EP inverse.This work extends corresponding results of the core-EP inverse.In Chapter 6,we focus on the perturbation bounds and continuity of the core-EP inverse and the DMP inverse of complex square matrices.Firstly,we investigate the continuity of the core-EP inverse without explicit error bounds,making use of rank equalities and matrix decomposition respectively.Secondly,the perturbation bounds of the core-EP inverse are derived in three reasonable cases,motivated by results for the perturbation bounds of the Drazin inverse.Finally,we derive an expression of the DMP inverse in terms of the Schur decomposition,and then the perturbation bounds and continuity of the DMP inverse are analyzed.